Faculty Publications
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Item Filtering in Time-Dependent Problems(Springer, 2023) Megha, P.; Godavarma, G.Spectral methods are efficient, robust and highly accurate methods in numerical analysis. When it comes to approximating a discontinuous function with spectral methods, it produces spurious oscillations at the point of discontinuity, which is called Gibbs’ phenomenon. Gibbs’ phenomenon reduces the spectral accuracy of the method globally. Filtering is a widely used method to prevent the oscillations due to Gibbs’ phenomenon by which the accuracy of the spectral methods is regained up to an extent. In this work, we study the effects of various filters in time-dependent problems and do a comparison of numerical results. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.Item A modified quasilinearization method for fractional differential equations and its applications(Elsevier Inc. usjcs@elsevier.com, 2015) Vijesh, V.; Roy, R.; Godavarma, G.Abstract In this paper, we prove an existence and uniqueness theorem for solving the nonlinear fractional differential equation of Caputo's type of order q ? (0, 1] using the method of modified quasilinearization. The main theorem has been illustrated numerically using appropriate examples which shows that the proposed quasilinearization method is robust and easy to apply. © 2015 Elsevier Inc.Item Extended convergence analysis of the newton-hermitian and skew-Hermitian splitting method(MDPI AG indexing@mdpi.com Postfach Basel CH-4005, 2019) Argyros, I.K.; George, S.; Godavarma, G.; Magreñán Ruiz, A.A.Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton-Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299-315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection-diffusion equations further validate the theoretical results. © 2019 by the authors.Item Iterative methods for a fractional-order Volterra population model(Rocky Mountain Mathematics Consortium PO Box 871804 Tempe AZ 85287-1804, 2019) Roy, R.; Vijesh, V.A.; Godavarma, G.We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the fractional-order integrodiffer- ential equation is solved numerically. The numerical experiments show that the proposed iterative scheme is more efficient than an existing iterative scheme in the literature, the convergence of which is very sensitive to various parameters, including the fractional order of the derivative. The spectral method based on our proposed iterative scheme shows greater flexibility with respect to various parameters. Sufficient conditions are provided to select the initial guess that ensures the quadratic convergence of the quasilinearization scheme. © 2019 Rocky Mountain Mathematics Consortium.Item Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations(Springer editorial@springerplus.com, 2020) Godavarma, G.; Prashanthi, K.S.; Vijesh, V.In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and uniqueness for fractional order initial value problem is given. This convergence analysis also proves quadratic convergence of the generalized quasilinearization method. Both the schemes are compared in terms of accuracy and convergence and it is found that IRBF scheme handles inherent RBF ill-condition better than corresponding DRBF method. Variety of numerical examples are provided and compared with other available results to confirm the efficiency of the schemes. © 2019, Springer Nature B.V.Item Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation(Springer Science and Business Media Deutschland GmbH, 2024) Revathy, J.M.; Godavarma, G.Radial basis function-based finite difference (RBF-FD) schemes generalize finite difference methods, providing flexibility in node distribution as well as the shape of the domain. In this paper, we consider a numerical formulation based on RBF-FD for solving a time–space fractional diffusion problem defined using a fractional Laplacian operator. The model problem is simplified into a local problem in space using the Caffarelli–Silvestre extension method. The space derivatives in the resulting problem are then discretized using a local RBF-based finite difference method, while L1 approximation is used for the fractional time derivative. Results obtained using the proposed scheme are then compared with that given in the existing literature. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Item On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion(Elsevier B.V., 2024) M, M.; Godavarma, G.; George, S.; Bate, I.; Senapati, K.Hueso et al. (2015) studied the fourth and sixth order methods to approximate a solution of a nonlinear equation in Rn, where the convergence analysis needs the involved operator to be five times differentiable and seven times differentiable for fourth-order and sixth-order methods, respectively. Also, they found no error estimate for those methods, as the Taylor series expansion played a leading role in proving the convergence. In this paper, we extended the method in the Banach space settings and relaxed the higher order derivative of the involved operator so that the methods can be used in a bigger class of problems which were not covered by the analysis in Hueso et al. (2015). Also, we obtained an error estimate without Taylor series expansion. This error estimate helps to get the number of iterations to achieve a given accuracy. Moreover, new sixth-order method is introduced by small modification and numerical examples were discussed for all these methods to validate our theoretical results and to study the dynamics. © 2024 Elsevier B.V.Item Unified convergence analysis of a class of iterative methods(Springer, 2025) M, M.; George, S.; Godavarma, G.In this paper, unified convergence analyses for a class of iterative methods of order three, five, and six are studied to solve the nonlinear systems in Banach space settings. Our analysis gives the number of iterations needed to achieve the given accuracy and the radius of the convergence ball precisely using weaker conditions on the involved operator. Various numerical examples have been taken to illustrate the proposed method, and the theoretical convergence has been validated via these examples. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.Item Enhancing the applicability of Jarratt-type fourth-order and sixth-order iterative methods(Springer Science and Business Media Deutschland GmbH, 2025) M, M.; Godavarma, G.; George, S.In this paper, we extended the applicability of the convergence analysis of the sixth-order iterative methods for solving nonlinear equations studied by Yaseen and Zafar (Arab J Math 11:585-599, 2022), whose analysis uses derivatives up to order seven. Also, we have done convergence analysis for the fourth-order method which can be obtained from their method by considering first two steps. Our analysis is applicable in more general Banach space settings and uses only the first three Frechet derivatives of the involved operator with Lipschitz-type conditions. Also, our analysis gives the computable radius of the convergence ball and the number of iterations to obtain the solution with a given accuracy. © The Author(s) 2025.Item Spectral error bound in the mollification of Fourier approximation using Gegenbauer polynomial based mollifier(Springer Nature, 2025) Megha, P.; Godavarma, G.Due to the global nature of the Fourier spectral methods, the Fourier approximation of discontinuous function gets impaired by spurious oscillations. This results in the approximation order reducing to and, respectively, for the discontinuous and non-discontinuous points. Nevertheless, it is shown by Gottlieb and others that higher order information is hidden in this approximation, using which spectral order accuracy can be recovered. Thus, in our work, we propose a spectral mollifier using the Gegenbauer polynomial kernel to regain the spectral accuracy of the Fourier approximation of a discontinuous function. Pointwise spectral accuracy has also been proved except for the discontinuous points. Results have been illustrated through examples. © The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics 2025.